2. Band Structure. Tight Bnding Approximation

This course consists of a general presentation of solar cells based on silicon thin films.
It is the third MOOC of the photovoltaic series of Ecole polytechnique on Coursera. The general aspects of the photovoltaic field are treated in "Photovoltaic Solar Energy". And the detailed description of the crystalline silicon solar cells can be found in "Physics of Silicon Solar Cells".
After a brief presentation of solar cells operation, thin film semiconductors are described here. The general properties of disordered and crystalline semiconductors are found very different, in particular in terms of band structure and doping mechanisms. Silicon thin films, generally less than 1 µm thick, are deposited from silane plasma leading to hydrogen incorporation. The growth mechanisms are discussed, in particular the capability to prepare partially crystallized thin films which appear as a mixture of nanocrystallites embedded in an amorphous tissue.
The consequences of the semiconductor properties on solar cells behavior are reviewed. The optical properties of amorphous and nanocrystalline silicon are complementary. Thus the plasma process is particularly well adapted to the preparation of multijunctions, with conversion efficiencies around 13-15 %. Furthermore plasma processes allow to prepare solar cells in large area on glass or flexible substrates.
Finally, it is shown that crystalline and amorphous silicon materials can be combine into heterojunctions solar cells with high efficiency conversion (about 25 %).
**This course is part of a series of 3**
Photovoltaic solar energy (https://www.coursera.org/learn/photovoltaic-solar-energy/)
Physics of silicon solar cells (https://www.coursera.org/learn/physics-silicon-solar-cells/)
Silicon thin film solar cells

教學方

Pr. Bernard Drevillon

Professor at Ecole polytechnique

Erik Johnson

CNRS Researcher

腳本

The band structure will first be investigated and we will introduce the tight binding model in order to understand why. So, these other semiconductors have a the band gap and also show that the gap is not a necessary consequence of the existence of the crystal order. So, let me remind you the qualitative approach that we used for the crystalline semiconductors. We were interested in two atoms that each address single electron around the nucleus. These two atoms were initially distance. When these atoms be close to were together, we obtained lift of degeneracy of the energy levels. The K shell is a core level and the L shell corresponds hazard to the valence band. This example illustrates the formation of the molecules, starting from the atoms. If we extend this analogy to N-body system with very large N, of the order of 10 to the 22 worked in energy bands with a continuum of states. This chemical approach does not assume any network periodicity. This periodicity of the network, may appear as a particular case. This can be shown more fundamentally from the tight binding approximations in the resolution of the Schrodinger equation, which is detailed in appendix one. Let us recall here the main points of this model without going into the details. The Schrodinger equation cannot be solved in the case of a very large number of atoms in a solid. The main hypothesis is to consider that the potential to reach an electron submitted in the solid, is only a weak perturbation as compared to that of the electron in the atom. In other words, the electron Hamiltonian in a solid, is not very different of that of the atom. The solid perturbation is approximated by your sum on the neighboring sites. The second hypothesis known as LCAO, linear combination of atomic orbitals, consists in assuming that the wave function solution of the Schrodinger equation, is a linear combination of atomic orbitals. In this approximation called tight binding, the periodicity of the potential is only a particular case. We thus obtain the energy level from the energy values of the Hamiltonian. This is what is indicated here, E of K. E of K therefore, equal to the atomic level Ei minus Alpha i, which shifts the atomic level by an intrasite constant. The term Beta I, corresponds to the overlap of the different orbitals. This intersite contributions linked to the overlap of the atomic orbitals, explain the electronic condition in the disorder material. This correspond to extradite state, that is to say, the possibility of K transport. Let us now turn to the sp3 hybridization. In the case of the elements of column four, diamond, carbon, silicon, or germanium, sp3 hybridization is discussed in detail in appendix two. Here we summarize the main aspects. The electronic configuration of the last shell should be s2p2. We would therefore expect a metallic structure since the states P form a distinct band, which is not totally filled, since only two of the six available states are filled. Actually, it is not the case. It is experimentally known that the silicon, whether crystalline or disorder is in the form of bands of electronic states separated by a band gap. We continue with qualitative study of the sp3 hybridization. We show in appendix two that forms of states S and P, one form sp3 atomic hybrid orbitals, which are linear combinations of the original S and P. Then the sp3 hybrids of neighboring sites combine and give rise to bonding and antibonding combinations. They give rise to energy bands separated by a band gap. The band corresponding to the bonding sp3 combination, is a valence band and the other is conduction band. They each comprise four atoms and therefore, the valence band is full at zero degree K, while the next band, the conduction band is empty at zero degree K. This applies to column four crystallite or disorder semiconductor. The sp3 hybridization and it's consequence, will be observed only if the energy cost of the hybridization is more than compensated by the energy gained due to the best overlap of the atomic orbitals. It can be seen in an appendix two, that it is greater than seven eV. The formation of energy bands of the elements of column four, can be summarized by the following animation. We start with orbital P and S, which combine by creating sp3 hybrids. These sp3 hybrids, then give rise to bonding and antibonding combinations which give rise to energy bands, valence and conduction bands. These properties appear as a consequence of local interactions without any hypothesis of periodicity. These energy bands are separated if the quantity Beta one, which corresponds to intersite contribution of equation 7.1, is sufficiently large and in particular greater than the energy difference Delta EPS between states P and state S. Beta one represents the integration between neighboring atoms hence the term tight binding. If the binding is sufficiently strong, then there will be creation of band gaps whatever the structure of the materials. However, in the case of amorphous, local fluctuation in potential can be expected. This fluctuation will influence the band edges so that there will be states in the band tails, linked to this fluctuation of potential. This animation illustrates the particular case of hydrogenated amorphous silicon. As before, states S and P with two electrons on its shell, create hybrids with four electrons. These hybrids then create bonding and antibonding combination. A dangling bond. That is to say, are lacking electron on a covalent bond, will thus create a defect which will correspond to the sp3 hybrid energy. It will be created in the middle of the band gap. This type of defect is a characteristic of a disorder material. These dangling bonds are taps which will be recommendation centers for electron-hole pairs. They'll pick your base, your operation of electronic devices such as solar cells. Finally, the band gap of amorphous silicon, is significantly higher than that of crystalline silicon. It corresponds approximately to the limit of the visible hunch. Let us mention some other consequences of the disorder. We've already pointed out the existence of band tails related to potential fluctuation. It can be shown that the band tails correspond to localized states and therefore, do not contribute to electrical condition. Electrical condition is in fact linked to extending states. We can therefore introduce the notion of mobility gap. Experimentally, the optical gap and the mobility gap are very close, with a different of the order of 0.1 to 0.2 electron volt. Another consequence of the disorder, is a lack of conservation of the vector K. This can be understood simply. A small disorder in a crystal, gives only a slight perturbation on the wave function, which corresponds to the diffusion between bloch states. On the other hand, the amorphous disorder can be very important. It can therefore induce a lot of coherence of a particularly interatomic distances. If one uses the Eisenberg relation, Delta P, Delta x, of the order of H bar. If A is atomic distance, we see the delta K. That is to say the uncertainty of a K, is in fact of the order of one over A, which is therefore of the order of K itself. In quantum mechanics, it means that K is not a good quantum number, since it cannot be determined precisely. Let's summarize the properties of disorder semiconductors in terms of density of energy states. The energy bands are no longer described by dispersion relations, A of K since K is no longer concerned. The distinction between direct gap on indirect gap thus disappears in amorphous materials. Amorphous semiconductors therefore, absorb much more than their crystalline counterparts, which is an advantage for photovoltaic applications. They can therefore be used in a thin film configuration. On the other hand, the disorder creates limitations of the electronic mobility, which can be easily understood from a small calculation of order of magnitude. I remind you that mobility Mu is defined by E2 over M, where two is the last time, L is the mean free path. Compared with crystalline silicon which has a mobility of the order 1,000 square centimeter per second on volt. If we estimate the order of magnitude of the disorder in an amorphous material by the interatomic distance, we see that the mobility of 1,000 for the crystal, is reduced to two or five for amorphous silicon. We shall see that in fact, it is about one square centimeter per volt on per second for electrons, in hydrogenated amorphous silicon. The description of the band structure of these other semiconductor I just been described it. I invite you to consult appendices one and two for a more detailed treatment. In the next session, we'll be interesting in the doping mechanisms of the semiconductor. Thank you.